On the role of gradient terms in quasilinear coercive differential inequalities on Carnot Groups

Abstract

In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $${\Delta }_{\mathbb{G}}^{\phi }u\ge b(x)f(u)l(|\nabla u|).$$ Prototype examples of ${\Delta }_{\mathbb{G}}^{\phi }$ are the (subelliptic) $p$-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient $l(t)$ that can vanish both as $t\to 0^{+}$ and as $t\to +\infty$. Our results improve on the recent literature and, by means of suitable counterexamples, we show that the range of parameters in the main theorems are sharp.

32 pages